# Full text of "Treatise On Analysis Vol-Ii"

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```13    MEASURES WITH BASE {JL        179

that the set of classes A, as A runs through £', is dense in I(X, //,). Show that h(u) =
h(u9 a) (Kolmogoroff-Sinai theorem). (Same method as in (a).)
(d) Take X to be the unit' circle U : |z| = 1 , let ju, be the image of Lebesgue measure
under the mapping t\—*e2nlt of [0, 1] onto U, and let u be the mapping zt-+e2niez.
Show that h(u) = 0. (Distinguish two cases according as 6 is rational or irrational.
In the latter case use (b), by taking a to be a partition into two half-open semicircles.)

11.    Let X, Y be two compact spaces and let (JL (resp. v) be a positive measure on X
(resp. Y) such that /x(X) = v(Y) = 1. A /^--measurable (resp. ^-measurable) mapping
u : X i— > X (resp. v : Y *—»• Y) such that U(JJL) = ju,(resp. v(v) = v) defines an endomorphism
U:f\-*(f°u)~  of L£(X, /^) (resp. an endomorphism V:g^(g°v)~ of Lc(Y, v))
(Section 13.11, Problem 10). The mappings u, v are said to be conjugate if there exists
an isometry T of I(X, ju) onto I(Y, v) (Problem 8) such that FQ r= T° U. Show
that if this is the case then h(u) — /z(y).

12.    Suppose that the measure ^ is bounded, and let p e [1, 4- oo[. Let (£/„) be a sequence
of continuous linear mappings of L&(X) into the space S(X, p,) (Problem 2). For any
function of fe JS?£ we denote by Un-fany function belonging to the class Un "/
and pu4

(tf!^ /)(*)== sup | (%•/)(*) |,

For each a > 0 let E«,N(/) denote the set of all x e X such that (US •/)(*) > a, and
let E«(/) denote the set of all x e X such that (U* •/)(*) > a.

(a)    Show that the set of /e L£ such that fi(Ea,N(/)) ^ e is closed, for each e > 0.

(b)    Assume that for all /e jgfg the function C7* -/is finite almost everywhere. Show
that under these conditions the number

C(a)=    sup    /x(E«(/))

Np(/)«l

tends to 0 as a -*• -f <x> (Banach's principle). (Use Baire's theorem in the complete
space L& .)

(c)    Under the hypotheses of (b), show that the set H of classes / such that the
sequence ((£/„ •/)(>;)) converges everywhere in X is closed in L&. (Write

remark that R(/)(*) = R(f—g)(x) almost everywhere for ff e H, and use (b).)

13. MEASURES WITH BASE p

(13.13.1)   Let g be a mapping ofX into C or into E. Then the following con-
ditions are equivalent:

(a)    For each x e X, there exists a neighborhood Vofx in X such that g<py
is integrable. the set of
```